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In recent years, there has been a flurry of activity towards proving lowerbounds for homogeneous depth-4 arithmetic circuits, which has brought us veryclose to statements that are known to imply $\textsf{VP} \neq \textsf{VNP}$. Itis open if these techniques can go beyond homogeneity, and in this paper wemake some progress in this direction by considering depth-4 circuits of lowalgebraic rank, which are a natural extension of homogeneous depth-4 circuits.A depth-4 circuit is a representation of an $N$-variate, degree-$n$ polynomial$P$ as \[ P = \sum_{i = 1}^T Q_{i1}\cdot Q_{i2}\cdot \cdots \cdot Q_{it} \; ,\] where the $Q_{ij}$ are given by their monomial expansion. Homogeneity addsthe constraint that for every $i \in [T]$, $\sum_{j} \operatorname{deg}(Q_{ij})= n$. We study an extension, where, for every $i \in [T]$, the algebraic rankof the set $\{Q_{i1}, Q_{i2}, \ldots ,Q_{it}\}$ of polynomials is at most someparameter $k$. Already for $k = n$, these circuits are a generalization of theclass of homogeneous depth-4 circuits, where in particular $t \leq n$ (andhence $k \leq n$).  We study lower bounds and polynomial identity tests for such circuits andprove the following results. We show an $\exp{(\Omega(\sqrt{n}\log N))}$ lowerbound for such circuits for an explicit $N$ variate degree $n$ polynomialfamily when $k \leq n$. We also show quasipolynomial hitting sets when thedegree of each $Q_{ij}$ and the $k$ are at most $\operatorname{poly}(\log n)$.  A key technical ingredient of the proofs, which may be of independentinterest, is a result which states that over any field of characteristic zero,up to a translation, every polynomial in a set of polynomials can be written asa function of the polynomials in a transcendence basis of the set. We combinethis with methods based on shifted partial derivatives to obtain our finalresults.

Arithmetic Circuits with Locally Low Algebraic Rank